For any tube \(t(\ell , r)\) ,
the projection
\(p_{\Pi } (t(\ell , r))\) 
is either a 2-ball of radius \(r\) ,
or a strip of width \(2r\) .
In either case,
\(p_{\Pi } (t(\ell , r))\) 
can be covered by a strip of width \(2r\) .
Hence the ball
\(B\)  of radius \(\varepsilon \) 
can be covered by strips
of total width less than \(2\varepsilon \) ;
this contradicts the
Plank Theorem
(for the planar case, see
English translation of
A. Tarski's and H. Moese's papers (1931–32)
in [1]} where the history
of the problem is also discussed;
the high-dimensional case is treated in [2]}).
